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/*******************************************************12Karl Rubin and I were wondering whether you could do a p-adic height3computation, the point of which is to figure out whether the p-adic4regulator of elliptic curves has a "tendency" to be not divisible by p. We5want to avoid a few degenerate cases, so the smallest p that is useful to6us is p=5. Is the strategy-- described below--, that systematically runs7through 5-adic height computations for points on quadratics twists of the8(non-3-Eisenstein) elliptic curve of conductor 37 reasonably easy to do?91011%%%%%%%%%%%%%%%%%%%12Our untwisted equation is E: y^2 = 4x^3-4x+1. For any nonzero integer D we13consider the twisted elliptic curve E_D: D.y^2 = 4x^3-4x+1. Put F(x): =144x^3-4x+1. Let x run through integers1516a= 0, \pm 1, \pm 2, \pm3, ...1718and for each of these a's,19201) Factor F(a) = d.b^2, where d is square-free;21222) throw out the ones where d is divisible by 5, or where the parity of the23rank of E_d is even;24253) for the rest, viewing P:=(a,b) as a rational point-- it will be of26infinite order--on the elliptic curve E_d, so one can compute the 5-adic27height of P on the elliptic curve E_d: this should be a 5-adic integer if28the height is normalized right, and there should be a good number of29instances where this height is a 5-adic unit; throw them out, and30314) print the rest.32%%%%%%%%%%%%%%%%3334Note that we only want to know the 5-adic height modulo 5.3536Then it would be our job to figure out whether strange things occur for37this printed list of instances: e.g., the point P might be divisible by 5,38or the Mordell-Weil rank might be > 2, or other stuff..., and we want to39know what other stuff there might be.4041****************************************************/424344